Design of fractional order controllers using constrained optimization and reference tracking method

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 11, No. 1, March 2020, pp. 291∼301
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v11.i1.pp291-301 r 291
Design of fractional order controllers using constrained
optimization and reference tracking method
Manoj D. Patil1
, K. Vadirajacharya2
, Swapnil Khubalkar3
1
Department of Electrical Engineering, Annasaheb Dange College of Engineering and Technology, Ashta, India
2
Department of Electrical Engineering, Dr. Babasaheb Ambedkar Technological University, Lonere, Raigad, India
3
Department of Electrical Engineering, G. H. Raisoni College of Engineering, Nagpur, India
Article Info
Article history:
Received Jun 7, 2019
Revised Jul 8, 2019
Accepted Nov 12, 2019
Keywords:
Constrained optimization
Controller
Fractional order control
Tuning
ABSTRACT
In recent times, fractional order controllers are gaining more interest. There are several
fractional order controllers are available in literature. Still, tuning of these controllers
is one of the main issues which the control community is facing. In this paper, online
tuning of five different fractional order controllers is discussed viz. tilted proportional-
integral-derivative (T-PID) controller, fractional order proportional-integral (FO-PI)
controller, fractional order proportional-derivative (FO-PD) controller, fractional
order proportional-integral-derivative (FO-PID) controller. A reference tracking
method is proposed for tuning of fractional order controllers. First order with dead
time (FOWDT) system is used to check feasibility of the control strategy.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Manoj D. Patil,
Assistant Professor, ADCET, Ashta, Maharashtra,
Tel: +91 9763530440
Email: mdpatileps@gmail.com
1. INTRODUCTION
The proportional-integral-derivative (PID) controllers are broadly used in many industrial
applications [1]. The simplicity of structure and easily implementable tuning strategies are behind their
success. Many industrial systems can be presented by first order with dead time (FOWDT) models and design of
controllers for such systems is important [2]. Ziegler-Nichols tuning, some nature inspired optimization
technique like genetic algorithm, particle swarm optimization, ant colony algorithm have widely used
techniques for PID controllers [3, 4]. Now-a-days, generalized form of PID controller is getting very
popular which is known as fractional order PID (FO-PID) controller [6]. In these controller strategies,
integral and differentiators are of fractional order. To use the fractional order is a step closure to real life
because the natural processes are mostly fractional [7, 8]. However, the fractional nature may be very small.
Clearly, the use of integer order in place of fractional order will differ significantly to the actual situation.
However, the integer order models are welcomed due to the absence of solutions to the fractional models.
Recently, the fractional order models are analyzed and implemented successfully in [9, 10]. It is shown that the
performance of conventional PID controllers can be improved using the extra degree of freedom from the use
of fractional integrator and differentiator [12, 13]. Fractional order controllers are being increasingly used for
controlling FOWDT systems [14].
Unlike traditional PID controller, the order of integration and/or differentiation in fractional
controllers are real and unknown parameters to be tuned. This leads to solving nonlinear equations to
design fractional controller. There are different optimization methods are tried to find the parameters of these
Journal homepage: http://ijpeds.iaescore.com

292 r ISSN: 2088-8694
controllers [15]. The conventional tuning rules are not applicable here. A genetic algorithm is proposed for
tuning in [16]. Particle swarm optimization is proposed in [17]. In [18], artificial bee colony algorithm is
proposed. Some other literature for tuning is reported in [19, 20]. Still, the area of tuning the fractional order
controllers is opened.
In [21], iterative feedback tuning algorithm is employed to tune a type of FO-PID controller.
The paper [22] deals with an optimal design of a new type variable coefficient FO-PID controller by using
heuristic optimization algorithms. Although many studies have tried to correct the performance of the system’s
transient and steady state responses. It is obvious that handling the step response and reference tracing together
will bring out a better control response. However, there are no studies using different controller parameters of
the system with uncertain plant structure. The major contribution of the paper is to fill this gap by presenting a
novel approach. In present paper, a reference tracking optimization method is proposed for tuning the fractional
controllers. Tuning of different fractional controllers is shown using this method. In the proposed method, even
though the controller has changed the optimization algorithm remains same. Only, the controller structure has
to be changed. If the plant gets changed, still the reference tracking optimization method works. Hence, the
proposed method can provide the solution for tuning of all other fractional controllers.
The organization of the paper is as: some preliminary information is given in Section II. Section III
introduces the different controller structures. In Section IV, the tuning methodology is proposed. In Section V,
simulation results are given. The paper has been concluded in Section VI.
2. PRELIMINARIES
Fractional calculus enables the use of real powers of the integration or differentiation operator.
The basics of fractional calculus are described in [23, 24]. Riemann-Liouville (RL) defined the fractional
operator as,
aDα
t f(t) =
1
Γ(m − α)
d
dt
!m Z t
a
f(τ)
(t − τ)α−m+1
dτ (1)
where, (m − 1) ≤ α < m, α is a real number, m is an integer, and Γ(.) is the Euler’s gamma function.
The Laplace transform of (2) gives,
L{aD±α
t f(t)} = s±α
F(s) (2)
s±α
has to be approximated in integer order for realization. There are several approximation techniques are
available [26].
2.1. Method of Approximation
Any transfer function can be approximated by interlacing real poles and zeros [27]. The integer order
approximation can be obtained within the desired band of frequency by using Oustaloup et al. method [24].
Here, poles and zeros are obtained as follows,
DN (s) =

ωu
ωh
α
ΠN
k=−N

1 + s/ω0
k
1 + s/ωk

(3)
where, ωk = ωb

ωh
ωb
(k+N+1/2+α/2)
(2N+1)

and ωk0 = ωb

ωh
ωb
k+N+1/2−α/2
(2N+1)

. The number of poles and zeros
are available as 2N+1, the approximation order is considered as N.
3. DESIGN OF FRACTIONAL ORDER CONTROLLERS
Various controller strategies, which are designed based on the fractional calculus, are discussed here.
3.1. Controller 1. T-PID
In [23], a different structure of fractional order controller is given, where the proportional term of
PID controller is changed with a tilted term with a transfer function s1/m
. The new transfer function of
T-PID controller closely approximate an optimal transfer function, hence provides improvement in controller.
The T-PID controller allows smaller effects of plant parameter variation and better disturbance rejection ratio
Int J Pow Elec Dri Syst, Vol. 11, No. 1, March 2020 : 291 – 301

Int J Pow Elec Dri Syst ISSN: 2088-8694 r 293
as compared to conventional PID controller. The T-PID controller is obtained as,
u(t) = KP D−α
e(t) + KI
Z
e(t) + KDDe(t) (4)
Applying Laplace transform,
u(s) = KP s−α
+ KI/s + KDs (5)
where, KP , KD, KI are proportional, derivative, and integral constant, u(s) is control signal, D is derivative
operator. The structure of T-PID controller is shown in Figure 1. It shows tilted component with proportional
component which is key factor in this controller structure.
Figure 1. Structure of T-PID controller
3.2. Controller 2. FO-PID
FO-PID controller used to minimize the error of the system. The error e(t) is scaled by gain KP in the
proportional control. The steady state error is minimized by integral control. The transients in the system are
taken care by derivative control. In FO-PID, in addition to KP , KD, and KI, two more parameters α - integrator
order, µ - order of differentiator are added to give more degree of freedom due to which improvement can be
seen in the controller’s performance. By adding up all these terms, FO-PID controller is achieved. The FO-PID
controller is given by,
u(t) = KP e(t) + KID−α
e(t) + KDDµ
e(t) (6)
Applying Laplace transform,
u(s) = KP + KIs−α
+ KDsµ
, (α, µ 0) (7)
When α = 1, µ = 1, PID controller is obtained. When α = 1, µ = 0, it becomes PI controller. At α = 0,
µ = 1, the PD controllers is obtained. The generalization of fractional controllers is shown in Figure 2. It shows
the different structures available in the PID domain. FO-PID can be anywhere in the plane. The structure of
FO-PID controller is depicted in Figure 3.
Figure 2. FO-PID generalization Figure 3. FO-PID controller
Design of fractional order controllers using... (Manoj D. Patil)

294 r ISSN: 2088-8694
3.3. Controller 3. FO-PI
When the µ = 0 and 2 α 0, then the FO-PI controller is obtained as,
u(s) = KP + KIs−α
(8)
The FO-PI controller is depicted in Figure 4. The difference is that derivative part is not present, here. If steady
state is key issue the FO-PI controller is the best choice.
3.4. Controller 4. FO-PD
When the α = 0 and 2 µ 0, then the FO-PD controller is obtained as,
u(s) = KP + KDsµ
(9)
Here, integral part is not in the controller structure. If the plant is having transients then this controller is best
choice. The FO-PD controller is depicted in Figure 5.
Figure 4. FO-PI controller
Figure 5. FO-PD controller
4. PROPOSED TUNING OF THE CONTROLLERS
Several techniques are available for optimizing the parameters of controllers. Here, constrained
optimization method with reference tracking is proposed to optimize the controller parameters. Constrained
minimization is the problem of finding a feasible solution that is optimum to a problem subject to constraints.
The aim of constrained optimization is to convert the problem into an smaller subproblems which can then
be solved by iterative methods. The characteristic of these methods is to translate the constrained problem to
unconstrained problem by using an objective function for constraints. In such a way, the solution is found out
for constrained problem using a sequence of unconstrained optimizations. The strategy of the simulated model
is shown in Figure 6. The constraints are kept on the plant response, including maximum overshoot, settling
time, rise time as shown in Figure 7. Also, the reference tracking signal is provided as a goal to be achieved as
shown in Figure 8. The model is simulated and the response is checked with initial parameters. The simulation
produces an un-optimized step response and the initial data for tuning. The plots are updated so that the design
requirements can satisfied.

Figure 6. Proposed scheme of optimization process of fractional controllers
Figure 7. Step response specifications Figure 8. Reference tracking specifications
The optimization algorithm are divided as large and medium scale. The large scale algorithm
requires linear algebra that does not store full matrices. This can be done by storing sparse matrices internally.
A large-scale algorithm can be used for a small problem. The medium-scale algorithm creates full matrices
internally and requires dense linear algebra. If a problem is significantly large, then full matrices require a large
amount of memory and a long time to execute. Whereas, a medium-scale algorithm provides extra functionality
such as additional constraint types which tends to provide better performance. The guidelines for constrained
optimization with reference tracking is as follows:
Step I: Start the process with the use of the ‘interior − point’ algorithm which is a large-scale
algorithm. ‘interior − point’ not only handles sparse large problems but also a small dense problems.
The algorithm satisfies constraints at all iterations, and can recover from ‘Inf/NaN’ results. After that go
for a different algorithms. The other algorithm may fail, as some algorithms may use more time and memory,
while some algorithms may not accept an initial point [28, 29]. If the problem did not solve then move towards
step II.
Step II: In this step, use sequential programming ‘sqp’ algorithm should be used. A ‘sqp’ algorithm
is one of the nonlinear programming methods. ‘sqp’ can satisfy bounds at all iterations. The algorithm can
recover easily from ‘Inf/NaN’. Here, a quadratic programming subproblem is solved at every major iteration.
This method allows constrained optimization [30]. An overview of ‘sqp’ is found in [31]. If the results are not
as per the requirement then go for Step III.
Step III: The ‘active − set’ algorithm can take large steps that gives speed to the algorithm.
The algorithm is effective for problems with nonsmooth constraints. In this method, active constraints are
included in canceling operation. These three steps will give the required results.
5. RESULTS AND DISCUSSION
Systems with first order plus dead time models have been classified in three categories based on
relative dead time δ which is function of time T and dead time τ as given below [32]:
1. Lag dominated systems - δ 0.1; where δ = τ
τ+T
2. Lag delay balanced systems - 0.1 δ 0.6
3. Delay dominated systems - δ 0.6
Here, the results are shown for lag dominated systems only. Whereas, the methodology is tested
against lag delay balanced and delay dominated system also.

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5.1. Case 1. with PID
The step response specification is shown in Figure 9 and the reference tracking specifications achieved
with PID controller is shown in Figure 10. Optimization progress with PID controller is shown in Table 1.
The obtained PID parameters are KP = 1.659, KI = 0.315, KD = 0.173.
Figure 9. Step response specifications with PID
controller
Time (seconds)
Amplitude
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 10. Reference tracking specifications
with PID controller
5.2. Case 2. with T-PID
Step response specification is shown in Figure 11 whereas reference tracking specifications achieved
with T-PID controller is depicted in Figure 16. Optimization progress with T-PID controller is shown in
Table 2. The obtained T-PID parameters are KP = 0.2, KI = 0.312, KD = 1.144, α = 0.8.
Table 1. Optimization Progress with PID Controller
Iteration
F-
count
Reference Tracking
Specifications
(minimum)
Step Response
Specification
(Upper) (=0)
Step Response
Specification
(Lower) (=0)
0 8 217.2816 -0.0396 -0.3833
1 32 51.6097 -0.0146 -0.4510
2 49 50.4433 -0.0142 -0.4093
3 57 14.5348 -0.0101 -0.2185
4 65 6.4976 -0.0093 0.0017
5 75 5.0369 -0.0089 0.0028
6 84 3.8441 -0.0090 0.0021
7 92 3.8441 -0.0090 0.0021
Figure 11. Step response specifications with T-PID
controller
Time (seconds)
Amplitude
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
with T-PID controller

Table 2. Optimization Progress with T-PID Controller
Iteration
F-
count
Reference Tracking
Specifications
(minimum)
Step Response
Specification
(Upper) (=0)
Step Response
Specification
(Lower) (=0)
0 8 325.7320 -0.0357 -0.0521
1 32 288.3060 -0.0394 -0.0493
2 46 248.7523 -0.0347 -0.0344
3 54 296.1709 -0.0135 -0.0015
4 62 177.7636 -0.0119 0.0018
5 76 124.0190 -0.0107 0.0062
6 94 123.9967 -0.0104 0.0076
7 104 103.6499 -0.0080 0.0105
8 112 71.6489 -0.0086 0.0103
9 121 66.5799 -0.0123 -0.0012
10 129 35.0891 -0.0113 0.0030
11 137 18.5853 -0.0076 0.0105
12 145 5.1998 -0.0104 0.0069
13 153 4.0939 -0.0103 0.0075
14 161 3.8244 -0.0104 0.0071
15 169 3.6999 -0.0103 0.0074
16 180 3.5236 -0.0101 0.0089
17 199 3.5099 -0.0100 0.0091
18 211 3.5039 -0.0100 0.0092
19 223 3.5039 -0.0100 0.0092
5.3. Case 3. with FO-PI
with FO-PI controller is depicted in Figure 14. Optimization progress with FO-PI controller is shown in
Table 3. The obtained FO-PI parameters are KP = 1.73, KI = 3.49, α = 0.55.
Figure 13. Step response specifications with FO-PI
controller
Time (seconds)
Amplitude
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
with FO-PI controller
Table 3. Optimization Progress with FO-PI Controller
Iteration
F-
count
Reference Tracking
Specifications
(minimum)
Step Response
Specification
(Upper) (=0)
Step Response
Specification
(Lower) (=0)
0 8 3.0496e+04 -0.4158 -0.4947
1 16 6.9294e+03 -0.1180 -0.2182
2 24 2.9033e+03 -0.0369 -0.1182
3 32 1.9057e+03 -0.0044 -0.0610
4 40 1.6598e+03 0.0014 -0.0253
5 53 1.1243e+03 -0.0248 -0.0173
6 65 1.1886e+03 -0.0220 -0.0135
7 76 1.4792e+03 0.0023 -0.0098
8 111 1.5002e+03 0.0020 -0.0096
9 145 1.5214e+03 0.0019 -0.0096
10 161 1.6484e+03 0.0013 -0.0090
11 183 1.6540e+03 0.0012 -0.0089
12 224 1.6540e+03 0.0012 -0.0089

298 r ISSN: 2088-8694
5.4. Case 4. with FO-PD
with FO-PD controller is shown in Figure 16. Optimization progress with FO-PD controller is shown in
Table 4. The obtained FO-PD parameters are KP = 107.8, KD = 1.968, µ = 0.63.
Figure 15. Step response specifications with FO-PD
controller
Time (seconds)
Amplitude
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
with FO-PD controller
Table 4. Optimization Progress with FO-PD Controller
Iteration
F-
count
Reference Tracking
Specifications
(minimum)
Step Response
Specification
(Upper) (=0)
Step Response
Specification
(Lower) (=0)
0 8 4.6123e+04 -0.5694 -0.6106
1 37 4.2841e+04 -0.5977 -0.5954
2 90 4.2834e+04 -0.5977 -0.5951
3 98 1.1651e+04 -0.1515 -0.2874
4 106 4.9829e+03 0.0738 -0.1398
5 114 2.3560e+03 0.0450 -0.0682
6 122 2.0831e+03 0.0241 -0.0318
7 130 2.1208e+03 0.0119 -0.0138
8 138 2.2145e+03 0.0049 -0.0051
9 146 2.3234e+03 0.0014 -0.0013
10 154 2.3443e+03 2.4580e-04 -1.3125e-04
11 162 2.3429e+03 2.4999e-05 -3.0894e-06
12 163 2.3429e+03 2.4999e-05 -3.0894e-06
5.5. Case 5. with FO-PID
with FO-PID controller is shown in Figure 18 . Optimization progress with FO-PID controller is shown in
Table 5. The obtained FO-PID parameters are KP = 107.8, KI = 0.05, KD = 73.9, α = 0.51, µ = 0.62.
Figure 17. Step response specifications with FO-PID
controller
Time (seconds)
Amplitude
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
with FO-PID controller

Table 5. Optimization Progress with FO-PID Controller
Iteration
F-
count
Reference Tracking
Specifications
(minimum)
Step Response
Specification
(Upper) (=0)
Step Response
Specification
(Lower) (=0)
0 8 3.3061e+04 -0.4414 -0.5273
1 16 8.4492e+03 -0.0965 -0.2408
2 24 3.6500e+03 0.0549 -0.1193
3 32 2.1838e+03 -0.0070 -0.0522
4 40 2.0712e+03 -0.0034 -0.0240
5 48 2.1917e+03 -0.0013 -0.0099
6 56 2.3034e+03 -6.4314e-04 -0.0033
7 64 2.3730e+03 -4.5363e-05 -6.6447e-04
8 72 2.3881e+03 1.2499e-06 -4.1323e-05
9 81 2.3873e+03 1.0911e-06 -1.2722e-05
10 90 2.3855e+03 1.2882e-06 -3.8773e-06
11 91 2.3855e+03 -3.8773e-06 1.2882e-06
Hence, all the available controllers can be tuned with the help of method discussed in this paper.
Only the controller structure need to be changed in the simulation and according to constrained and reference
tracking method every type of fractional order controller can be optimized. All the five controllers are compared
with each other and FO-PID controller is found to be superior in all the five cases. Since, the FO-PID controller
is the combination of fractional integrator and differentiator. Fractional differentiator takes care of transient
response while fractional integrator looks after the steady state response.
6. CONCLUSION
In this paper, different structures of FO-PID controllers are designed and applied to FOWDT process.
The tuning methods of fractional controllers involves complex equations. So, it is difficult to find a solution to
the problem. Because of that, the tuning method is simplified so that the controller can be tuned very easily and
effectively. So that, the constrained optimization algorithm with reference tracking specifications is found to
provide solution to tune these controller strategies. The proposed tuning algorithm can be applied to different
control strategies and different plant structures.
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BIOGRAPHIES OF AUTHORS
Mr. Manoj D. Patil was born in Sangli, Maharashtra, India, in 1987. He has received his B.E.
Degree in Electrical Engineering from Shivaji University Kolhapur, Maharashtra, India in 2009, and
the M.E. Degree in Electrical Power Systems from Government College of Engineering Aurangabad
(which is affiliated to Dr. Babasaheb Ambedkar Marathwada University Aurangabad), Maharashtra,
India in 2011. He is working as Assistant Professor at Annasaheb Dange College of Engineering
Technology, Ashta, Sangli, Maharashtra since July 2011. He is currently working toward the
Ph.D. degree with the Division of Electrical Engineering at Dr. Babasaheb Ambedkar Technological
University (BATU), Lonere, Raigad, Maharashtra, India.

Dr. K. Vadirajacharya received the B.E. degree in Electrical Engineering from PDA College of Engi-
neering, Gulbarga, India, in 1984, the M.E. degree in Power Electronics from GSITS Indore, India, in
1994, and the Ph.D. degree from the Indian Institute of Technology Roorkee, India, in 2009. He has
more than 25 publications in different national and international conferences and journals of repute.
Currently, he is an Associate Professor in the Department of Electrical Engineering, BATU, Lonere,
India. His areas of interest include power quality conditioning, flexible ac transmission device, energy
audit, and energy conservation.
Dr. Swapnil Khubalkar received the B.E. degree in Electronics Engineering in 2010, the M.Tech.
degree in Industrial Drives and Control in 2012 from Nagpur University. He received the Ph.D.
degree from the Visvesvaraya National Institute of Technology, Nagpur, India, in 2018. He has
more than 15 publications in different national and international conferences and journals of repute.
Currently, he is an Assistant Professor in the Department of Electrical Engineering, G.H. Raisoni
College of Engineering, Nagpur. His areas of interest include Power Electronics, Drives, Fractional
order control, etc.

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